# Heat equation

1. Oct 18, 2009

### Raven2816

Problem:
u (sub t) = (1/2)u (sub xx)

find the solution u(x,t) of the heat equation for the following initial conditions:

u(x,0) = x
u(x,0) = x^2
u(x,0) = sinx
u(x,0) = 0 for x < 0 and 1 for x>=0

i'm really flying blind here. I've taken differential equations years ago but nothing is too familiar. i know this is second order and that's really confusing me.

so for the x^2 condition i've tried differentiating up to 3 times and simplifying. i got a solution: x^2 + t. i got it by accident so it probably isn't right.
i feel like since there are no boundaries i should be able to integrate both sides, and the plug in my initial conditions but i'm just confused in general. everything i look up online has boundaries so i'm struggling to find a comparable example to learn from.

any tips or advice would be a great help.

2. Oct 18, 2009

### Mapes

This problem is sometimes called "diffusion on the whole line." It's covered in most texts on partial differential equations. It's not worth it to attack it by trial and error.

3. Oct 20, 2009

### gabbagabbahey

The easiest way to approach this problem is probably to use the method of "Separation of Variables". That is; assume the solution is of the form $u(x,t)=X(x)T(t)$, substitute this assumed form into your PDE and solve the resulting ODEs.